L(s) = 1 | + (1.5 + 0.866i)3-s + (1 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−1 − 1.73i)13-s − 3·17-s + 19-s + (3 − 1.73i)21-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + 5.19i·27-s + (−3 + 5.19i)29-s + (−2 − 3.46i)31-s + (−4.5 + 2.59i)33-s − 4·37-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (0.377 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.277 − 0.480i)13-s − 0.727·17-s + 0.229·19-s + (0.654 − 0.377i)21-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + 0.999i·27-s + (−0.557 + 0.964i)29-s + (−0.359 − 0.622i)31-s + (−0.783 + 0.452i)33-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37906 + 0.243166i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37906 + 0.243166i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.29412028177970020405968736759, −12.32959609175909797451294911946, −10.74238199510750267337032779634, −10.20755662618736127992191657134, −8.985259579193792400133529119334, −7.957456630593232112232971517021, −7.00489071735715296868076371624, −5.05711936670991334027722453981, −4.00947033884517203424427915554, −2.35108135638940597967821092355,
2.04901078629379074055897362975, 3.53519034210908129381929486333, 5.31075452569347015793858506264, 6.72808141967301058242909305741, 7.914504181237864992656505890458, 8.773992186911460946511155514314, 9.699742114448970930952488412869, 11.21134912092071582860274237943, 12.08741449551169059831330125042, 13.26753273033277262417649451966